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Full text of "Mathematical And Physical Papers - Iii"

ON  THE MOTION  OF PENDULUMS.
material change would thus be made in the numerical result, , may be worth while to point out the mode of solution of the problem. Suppose the globule preserved in a strictly spherical shape by capillary attraction, which will very nearly indeed be the case. Conceive a velocity equal and opposite to that of the globule impressed both on the globule and on the surrounding fluid, which will reduce the problem to one of steady motion. Let ^, &c. refer to the fluid forming the globule, and assume <& =ft (r) sin2 9. Then we get on changing the constants in (119)
 (r) = Af* + S,r + Of + D,r\
The arbitrary constants, A19 Bl vanish by the condition that the velocity shall not become infinite at the centre. There remain the two arbitrary constants GI} Dl to be determined, in addition to those which appeared in the former problem. But we have now four instead of two equations of condition which have to be satisfied at the surface of the sphere, which are that
5=0,    ^ = 0,     = ,,    Te = Tie,   whenr = a    ......(128).
We shall thus have the same number of arbitrary constants as conditions to be satisfied. Now Tld will involve ^ as a coefficient, just as TQ involves p!p or p,; and //,i; which refers to water, is much larger than yL6, which refers to air, although p! is larger than ^. Hence the results will be nearly the same as if we had taken ^ = 00, or regarded the sphere as solid.
If, however, instead of a globule of liquid descending in a gas we have a very small bubble ascending in a liquid, we must not treat the bubble as a solid sphere. We may in this case also neglect the motion of the fluid forming the sphere, but we have now arrived at the other extreme case of the general problem, and the two equations of condition which have to be satisfied fit the surface of the sphere are that R = 0 and Te = 0 when r = a, instead of R = 0 and 0 = 0, when r = a.
The equation of condition Te = 0 which applies to a bubble, as well as the fourth of equations (128), will not bo the true equations, if forces arising from internal friction exist in the superficial film of a fluid which are of a different order of magnitude from those which exist throughout the mass. At the end of the memoir already referred to, Coulomb states that in very slow motions the resistance of bodies not completely immersed in a liquid is much                         !'
greater than that of bodies wholly immersed,   and   promises   to